I called several child custody evaluators from the list I got. Most of them don't even do the work anymore, or have moved away.
I just talked to one who has a very distinguished reputation. He was mentioned by a couple of the others I called. His first question was who my lawyer was. When I told him I didn't have one, he said that he does not do cases anymore where one or both parties is "pro per" (meaning unrepresented). He offered to explain why, and told me that the lawyers help manage the evaluation.
I told him that I found that surprising, because in my experience, the lawyers are not supposed to have anything to do with the evaluation. They are not even allowed to talk to the evaluator. He said that was true, but he had a couple of recent cases in which the lawyer dropped out, and it created extra trouble for him somehow. He didn't explain how, but said that lawyers allow him to focus on the evaluation, especially in difficult cases.
Now I am wondering just what the lawyers do to make the evaluator's life easier, when the lawyers are not supposed to be even involved. Do the lawyers make sure that the bills are paid? Do the lawyers somehow browbeat the clients into cooperating with whatever the evaluator wants to do? Do the lawyers secretly give some info to the evaluators?
At any rate, the claim that lawyers are not involved seems to be false. The lawyers are doing something. Exactly what, I don't know, but they are doing something outside of what the rules allow.